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On transposed Poisson conformal algebras

Lamei Yuan, Hao Fang

Abstract

The aim of this paper is to introduce the notion of (noncommutative) transposed Poisson conformal algebras, which serve as the conformal analogues of transposed Poisson algebras and admit a rich class of identities. We show that the tensor product of two transposed Poisson conformal algebras is also a transposed Poisson conformal algebra. Moreover, we establish a close relationship between transposed Poisson conformal algebras and Hom-Lie conformal algebras, and give the compatibility conditions between a Poisson conformal algebra and a transposed Poisson conformal algebra. In addition, we provide several constructions of transposed Poisson conformal algebras arising from related algebraic structures. Finally, a complete classification of compatible noncommutative transposed Poisson conformal algebraic structures over a class of Lie conformal algebras W(a, b) is given.

On transposed Poisson conformal algebras

Abstract

The aim of this paper is to introduce the notion of (noncommutative) transposed Poisson conformal algebras, which serve as the conformal analogues of transposed Poisson algebras and admit a rich class of identities. We show that the tensor product of two transposed Poisson conformal algebras is also a transposed Poisson conformal algebra. Moreover, we establish a close relationship between transposed Poisson conformal algebras and Hom-Lie conformal algebras, and give the compatibility conditions between a Poisson conformal algebra and a transposed Poisson conformal algebra. In addition, we provide several constructions of transposed Poisson conformal algebras arising from related algebraic structures. Finally, a complete classification of compatible noncommutative transposed Poisson conformal algebraic structures over a class of Lie conformal algebras W(a, b) is given.
Paper Structure (10 sections, 17 theorems, 118 equations)

This paper contains 10 sections, 17 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transposed Poisson conformal algebras
  4. Definition of transposed Poisson conformal algebras
  5. Identities in transposed Poisson conformal algebras
  6. Tensor products of transposed Poisson conformal algebras
  7. Relation with Hom-Lie conformal algebras
  8. Compatibility conditions between PCA and TPCA
  9. Constructions of transposed Poisson conformal algebras
  10. Compatible noncommutative TPCA structures on the Lie conformal algebra $W(a,b)$

Key Result

Proposition 2.10

Associative conformal algebras are left-symmetric conformal algebras. If $(A,\ast_\lambda)$ is a left-symmetric conformal algebra, then the $\lambda$-bracket defines a Lie conformal algebra $\mathfrak{g}(A)$, which is called a sub-adjacent Lie conformal algebra of $A$.

Theorems & Definitions (51)

  • Definition 2.1: Kac99
  • Definition 2.2: Kac98
  • Remark 2.3: DAK98
  • Remark 2.4: KN23
  • Definition 2.5: BKV99
  • Example 2.6: DAK98
  • Example 2.7: LHW20
  • Definition 2.8: Yuan14
  • Definition 2.9: Kac98
  • Proposition 2.10: HL15
  • ...and 41 more